Generalizations of theorems from the Theory of Functions

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Text from introduction: "Practically all of the text books and lectures on the
Theory of Functions of a Complex Variable treat only those functions
which have a derivative in every point of some given region.
This derivative must be independent of the direction in
which the increment of the variable is taken. Functions which
have this property are called analytic functions of a complex
variable. In some cases these are called simply functions, the
non analytic functions not being considered functions at all."